Convolution identity involving the Möbius function $\sum_{d|n,d>0} |\mu(d)| = 2^{\omega(n)}$

Question

I'm learning about the Möbius Inversion Formula but I'm stuck on an exercise which involves the Möbius function.

Let $n\in\mathbb{Z}$ with $n>0$ and let $\omega(n)$ denote the number of distinct prime numbers dividing n. Prove that

$$\sum_{d|n,d>0} |\mu(d)| = 2^{\omega(n)}$$

I can see that $\sum|\mu(d)|$ is divisible by $2$ but I don't know how to relate it to $\omega(n)$. Could someone give me a hint in the right direction on how to approach this proof?

Answer 1

Hint: Suppose that $P = \{p_1,\ldots,p_{\omega(n)}\}$ are the distinct primes divising $n$. Then $\mu(d) \neq 0$ (and so $|\mu(d)| = 1$) for $d|n$ only if $d$ is a product of a subset of $P$.

Answer 2

(Brutal solution: )

$|\mu(n)|$ is also a Arithmetic function, so is $f(n) = \sum_{d|n}|\mu(d)|$.

Since $f(p^n) = 2$

Then $f(p^aq^b \dots) = f(p^a)f(q^b) \dots = 2^{\omega(n)}$